The Goddard High Resolution Spectrograph aboard the Hubble Space Telescope was used with the G160M grating to obtain high-resolution (6.2 A) spectra of three very bright active galactic nuclei located behind voids in the nearby distribution of bright galaxies (i.e., CfA and Arecibo redshift survey regions). A total of eight definite (greater than or equal to 4 sigma) Ly(alpha) absorption lines were discovered ranging in equivalent width from 26 to 240 mA at Galactocentric velocities 1740-7740 km/s. Of these eight systems, we locate seven in supercluster structures and one, in the sight line of Mrk 501 at 7740 km/s, in a void. In addition, one of two tentative (3-4 sigma) Ly(alpha) absorption lines are found in voids. Thus, the voids are not entirely devoid of matter, and not all Ly(alpha) clouds are associated with galaxies. Also, since the path lengths through voids and superclusters probed by our observations thus far are nearly equal, there is some statistical evidence that the Ly(alpha) clouds avoid the voids. The nearest galaxy neighbors to these absorbing clouds are 0.45-5.9 Mpc away, too far to be physically associated by most models. The lower equivalent width absorption lines (W(sub lambda) less than or equal to 100 mA) are consistent with random locations with respect to galaxies and may be truly intergalactic, similar to the bulk of the Ly(alpha) forest seen at high z. These results on local Ly(alpha) clouds are in full agreement with those found by Morris et al. (1993) for the 3C 273 sight line but are different from the results for higher equivalent width systems where closer cloud-galaxy associations were found by Lanzetta et al. (1994). Pencil-beam optical and 21 cm radio line observations of the area of sky surrounding Mrk 501 fail to find faint galaxies near the velocities of the Ly(alpha) clouds in that sight line. Specifically, for the 'void absorption' system at 7740 km/s, we find no galaxy at comparable redshift to the absorber within 100 h(sub 75)(sup -1) kpc (H(sub 0) = 75 h(sub 75) km/s Mpc(sup -1)) with an absolute magnitude of B less than or equal to - 16 and no object with H I mass greater than or equal to 7 x 10(exp 8) h(sub 75)(sup -2) M(solar) within 500 h(sub 75)(sup -1) kpc. Thus, neither a faint optical galaxy nor a gas-rich, optically dim or low surface brightness galaxy is present close to this absorber.